@Prove It. I am reading again this post. And, sorry, I have something to ask again. I can add something more:
If we set:

$5^{2(k+1)}-1 = 24p$

from the following passage:

$24 \cdot 5^{2k} + 5^{2k} - 1 = 24p$

we have, that the following quantity is divisible by 24, for inductive hypotesis:

$5^{2k} - 1 = 24m$

and also the following quantity is divisible by 24:

$24 \cdot 5^{2k} = 24s$

therefore, the inductive step, became a sum of two positions

$24p = 24m + 24s$
$24p = 24(m+s)$
$p = m+s$

So the desired value for the integer p is m+s.
But, what I want to ask is:
How can I be sure about the rightness of the result we have found?
Maybe, Do I have to substitute that $p$ in the $P(n)$, and try to find something useful like this?